BS2 - Proving Goldbach's Conjecture Using Bertrand's Postulate
SCURS Disciplines
Mathematics
Document Type
General Presentation (Oral)
Invited Presentation Choice
Not Applicable
Abstract
Goldbach’s Conjecture asserts that every even integer greater than 2 can be ex- pressed as the sum of two primes. Despite extensive computational verification, a general proof remains unknown. Motivated by Bertrand’s Postulate, we examine the distribution of primes within the consecutive intervals [⌊ x/2 ⌋, x) and (x, ⌊ 3x/2 ⌋]. For each prime p ∈ [⌊ x/2 ⌋, x), consider its reflection across x, namely q = 2x − p, which lies in x, ⌊ 3x/2 ⌋]. We conjecture that for every integer x ≥ 4, there exists at least one prime p ∈ [⌊ x/2 ⌋, x) such that its reflected counterpart q is also prime. Such a pair satisfies p + q = 2x, yielding a symmetric prime decomposition of the even integer 2x. If established, this symmetry principle would imply Goldbach’s Conjecture for all even integers ≥ 8. Computational experiments for primes up to 10^6 provide supporting evidence for the conjectured phenomenon. It must be noted that there are some anomalous x values which do not yield any prime q values without a minor adjustment to the algorithm. When x = 19, for example, it has (p, q) pairings of (11, 27), (13, 25), and (17, 21). As we can see, none of the q values are prime. However, if we extend the left-hand interval to include 7, doing so does not violate Bertrand, and we get the pairing of (7, 31), thereby preserving the claim of prime symmetry.
Keywords
Prime Numbers, Goldbach’s Conjecture, Bertrand’s Postulate, Symmetry of Primes, Equidistant Primes
Start Date
10-4-2026 2:25 PM
Location
CASB 102
End Date
10-4-2026 2:40 PM
BS2 - Proving Goldbach's Conjecture Using Bertrand's Postulate
CASB 102
Goldbach’s Conjecture asserts that every even integer greater than 2 can be ex- pressed as the sum of two primes. Despite extensive computational verification, a general proof remains unknown. Motivated by Bertrand’s Postulate, we examine the distribution of primes within the consecutive intervals [⌊ x/2 ⌋, x) and (x, ⌊ 3x/2 ⌋]. For each prime p ∈ [⌊ x/2 ⌋, x), consider its reflection across x, namely q = 2x − p, which lies in x, ⌊ 3x/2 ⌋]. We conjecture that for every integer x ≥ 4, there exists at least one prime p ∈ [⌊ x/2 ⌋, x) such that its reflected counterpart q is also prime. Such a pair satisfies p + q = 2x, yielding a symmetric prime decomposition of the even integer 2x. If established, this symmetry principle would imply Goldbach’s Conjecture for all even integers ≥ 8. Computational experiments for primes up to 10^6 provide supporting evidence for the conjectured phenomenon. It must be noted that there are some anomalous x values which do not yield any prime q values without a minor adjustment to the algorithm. When x = 19, for example, it has (p, q) pairings of (11, 27), (13, 25), and (17, 21). As we can see, none of the q values are prime. However, if we extend the left-hand interval to include 7, doing so does not violate Bertrand, and we get the pairing of (7, 31), thereby preserving the claim of prime symmetry.